A Stronger Gordon Conjecture and an Analysis of Free Bicuspid Manifolds with Small Cusps
2018
Thurston showed that for all but a finite number of
Dehn Surgeries on a cusped hyperbolic 3-manifold, the resulting
manifold admits a hyperbolic structure. Global bounds on this
number have been set, and gradually improved upon, by a number of
Mathematicians until Lackenby and Meyerhoff proved the sharp bound
of 10, which is realized by the figure-eight knot exterior. We
improve this result by proving a stronger version of Gordon’s
conjecture: that excluding the figure-eight knot exterior, cusped
hyperbolic 3-manifolds have at most 8 non-hyperbolic Dehn
Surgeries. To do so we make use of the work of Gabai et. al. from a
forthcoming paper which parameterizes measurements of the cusp,
then uses a rigorous computer aided search of the space to classify
all hyperbolic 3-manifolds up to a specified cusp size. Their
approach hinges on the discreteness of manifold points in the
parameter space, an assumption which cannot be made if the
manifolds have infinite volume. In this paper we also show that
infinite-volume manifolds, which must be Free Bicuspid, can have
cusp volume as low as 3.159. As such, these manifolds are a concern
for any future expansion of the approach of Gabai et.
al.
- Correction
- Source
- Cite
- Save
- Machine Reading By IdeaReader
0
References
1
Citations
NaN
KQI